Introduction to Financial Econometrics Hypothesis Testing in the Market Model

Introduction to Financial Econometrics Hypothesis Testing in the Market Model

Introduction to Financial Econometrics

Hypothesis Testing in the Market Model

Eric Zivot Department of Economics University of Washington February 29, 2000

1 Hypothesis Testing in the Market Model

In this chapter, we illustrate how to carry out some simple hypothesis tests concerning the parameters of the excess returns market model regression

To be completed

Using the market model regression,

Rt = α + βRM t+ εt, t = 1, , T

εt ∼ iid N(0, σ2ε), εt is independent of RM t (1) consider testing the null or maintained hypothesis α = 0 against the alternative that

α6= 0

H0 : α = 0 vs H1 : α6= 0

If H0 is true then the market model regression becomes

Rt= βRM t+ εt

and E[Rt|RM t = rM t] = βrMt We will reject the null hypothesis, H0 : α = 0, if the estimated value of α is either much larger than zero or much smaller than zero Assuming H0 : α = 0 is true, ˆα ∼ N(0, SE(ˆα)2) and so is fairly unlikely that ˆα will

be more than 2 values of SE(ˆα) from zero To determine how big the estimated value

of α needs to be in order to reject the null hypothesis we use the t-statistic

tα=0= αb − 0

d

SE(α)b ,

whereα is the least squares estimate of α andb SE(d α) is its estimated standard error.b

The value of the t-statistic, tα=0, gives the number of estimated standard errors that

b

α is from zero If the absolute value of tα=0 is much larger than 2 then the data cast considerable doubt on the null hypothesis α = 0 whereas if it is less than 2 the data are in support of the null hypothesis1 To determine how big | tα=0| needs to be to reject the null, we use the fact that under the statistical assumptions of the market model and assuming the null hypothesis is true

tα=0 ∼ Student − t with T − 2 degrees of freedom

If we set the significance level (the probability that we reject the null given that the null is true) of our test at, say, 5% then our decision rule is

Reject H0 : α = 0 at the 5% level if |tα=0| > tT −2(0.025) where tT −2 is the 212% critical value from a Student-t distribution with T− 2 degrees

of freedom

Example 1 Market Model Regression for IBM

Consider the estimated MM regression equation for IBM using monthly data from January 1978 through December 1982:

b

RIBM,t=−0.0002

(0.0068) + 0.3390

(0.0888) ·RM t, R2 = 0.20, σb ε = 0.0524 where the estimated standard errors are in parentheses Here α =b −0.0002, which is very close to zero, and the estimated standard error, SE(ˆd α) = 0.0068, is much larger

than α The t-statistic for testing Hb 0 : α = 0 vs H1 : α6= 0 is

tα=0 = −0.0002 − 0

0.0068 =−0.0363

so thatα is only 0.0363 estimated standard errors from zero Using a 5% significanceb

level, t58(0.025)≈ 2 and

|tα=0| = 0.0363 < 2

so we do not reject H0 : α = 0 at the 5% level

1 This interpretation of the t-statistic relies on the fact that, assuming the null hypothesis is true

so that α = 0, b α is normally distributed with mean 0 and estimated variance d SE( b α) 2

1.3 Testing Hypotheses about β

In the market model regression β measures the contribution of an asset to the vari-ability of the market index portfolio One hypothesis of interest is to test if the asset has the same level of risk as the market index against the alternative that the risk is different from the market:

H0 : β = 1 vs H1 : β6= 1

The data cast doubt on this hypothesis if the estimated value of β is much different from one This hypothesis can be tested using the t-statistic

tβ=1 = βb − 1

d

SE(β)b

which measures how many estimated standard errors the least squares estimate of β

is from one The null hypothesis is reject at the 5% level, say, if|tβ=1| > tT −2(0.025) Notice that this is a two-sided test

Alternatively, one might want to test the hypothesis that the risk of an asset is strictly less than the risk of the market index against the alternative that the risk is greater than or equal to that of the market:

H0 : β = 1 vs H1 : β≥ 1

Notice that this is a one-sided test We will reject the null hypothesis only if the estimated value of β much greater than one The t-statistic for testing this null hypothesis is the same as before but the decision rule is different Now we reject the null at the 5% level if

tβ=1 <−tT −2(0.05) where tT −2(0.05) is the one-sided 5% critical value of the Student-t distribution with

T − 2 degrees of freedom

Example 2 MM Regression for IBM cont’d

Continuing with the previous example, consider testing H0 : β = 1 vs H1 : β 6= 1 Notice that the estimated value of β is 0.3390, with an estimated standard error of 0.0888, and is fairly far from the hypothesized value β = 1 The t-statistic for testing

β = 1 is

tβ=1 = 0.3390− 1

0.0888 =−7.444 which tells us that β is more than 7 estimated standard errors below one Sinceb

t0.025,58 ≈ 2 we easily reject the hypothesis that β = 1

Now consider testing H0 : β = 1 vs H1 : β ≥ 1 The t-statistic is still -7.444 but the critical value used for the test is now −t58(0.05) ≈ −1.671 Clearly, tβ=1 =

−7.444 < −1.671 = −t58(0.05) so we reject this hypothesis

1.4 Testing Joint Hypotheses about α and β

Often it is of interest to formulate hypothesis tests that involve both α and β For example, consider the joint hypothesis that α = 0 and β = 1 :

H0 : α = 0 and β = 1

The null will be rejected if either α 6= 0, β 6= 1 or both Thus the alternative is formulated as

H1 : α6= 0, or β 6= 1 or α 6= 0 and β 6= 1

This type of joint hypothesis is easily tested using a so-called F-test The idea behind the F-test is to estimate the model imposing the restrictions specified under the null hypothesis and compare the fit of the restricted model to the fit of the model with

no restrictions imposed

The fit of the unrestricted (UR) excess return market model is measured by the (unrestricted) sum of squared residuals (RSS)

SSRU R= SSR(ˆα, ˆβ) =

T

X

t=1

b

ε2t =

T

X

t=1

(Rt−αb −βRb M t)2

Recall, this is the quantity that is minimized during the least squares algorithm Now, the market model imposing the restrictions under H0 is

Rt = 0 + 1· (RM t− rf) + εt

= RMt+ εt Notice that there are no parameters to be estimated in this model which can be seen

by subtracting RMt from both sides of the restricted model to give

Rt− RM t =eεt

The fit of the restricted (R) model is then measured by the restricted sum of squared residuals

SSRR= SSR(α = 0, β = 1) =

T

X

t=1

e

ε2t =

T

X

t=1

(Rt− RM t)2 Now since the least squares algorithm works to minimize SSR, the restricted error sum of squares, SSRR, must be at least as big as the unrestricted error sum of squares, SSRU R If the restrictions imposed under the null are true then SSRR ≈ SSRU R

(with SSRR always slightly bigger than SSRU R) but if the restrictions are not true then SSRR will be quite a bit bigger than SSRU R The F-statistic measures the (adjusted) percentage difference in fit between the restricted and unrestricted models and is given by

F = (SSRR− SSRU R)/q

SSRU R/(T − k) =

(SSRR− SSRU R)

q·σb2ε,U R ,

where q equals the number of restrictions imposed under the null hypothesis, k denotes the number of regression coefficients estimated under the unrestricted model and

b

σ2ε,UR denotes the estimated variance of εt under the unrestricted model Under the assumption that the null hypothesis is true, the F-statistic is distributed as an F random variable with q and T − 2 degrees of freedom:

F ∼ Fq,T −2 Notice that an F random variable is always positive since SSRR> SSRUR The null hypothesis is rejected, say at the 5% significance level, if

F > Fq,T −k(0.05) where Fq,T −k(0.05) is the 95% quantile of the distribution of Fq,T −k

For the hypothesis H0 : α = 0 and β = 1 there are q = 2 restrictions under the null and k = 2 regression coefficients estimated under the unrestricted model The F-statistic is then

Fα=0,β=1 = (SSRR− SSRU R)/2

SSRU R/(T − 2) Example 3 MM Regression for IBM cont’d

Consider testing the hypothesis H0 : α = 0 and β = 1 for the IBM data The unrestricted error sum of squares, SSRU R, is obtained from the unrestricted regression output in figure 2 and is called Sum Square Resid:

SSRU R = 0.159180

To form the restricted sum of squared residuals, we create the new variable eεt =

Rt − RM t and form the sum of squares SSRR = P T

t=1 eε2t = 0.31476 Notice that SSRR> SSRU R The F-statistic is then

Fα=0,β=1= (0.31476− 0.159180)/2

0.159180/58 = 28.34.

The 95% quantile of the F-distribution with 2 and 58 degrees of freedom is about 3.15 Since Fα=0,β=1 = 28.34 > 3.15 = F2,58(0.05) we reject H0 : α = 0 and β = 1 at the 5% level

In many applications of the MM, α and β are estimated using past data and the estimated values of α and β are used to make decision about asset allocation and risk over some future time period In order for this analysis to be useful, it is assumed that the unknown values of α and β are constant over time Since the risk characteristics of

assets may change over time it is of interest to know if α and β change over time To illustrate, suppose we have a ten year sample of monthly data (T = 120) on returns that we split into two five year subsamples Denote the first five years as t = 1, , TB

and the second five years as t = TB+1, , T The date t = TB is the “break date” of the sample and it is chosen arbitrarily in this context Since the samples are of equal size (although they do not have to be) T− TB = TB or T = 2· TB The market model regression which assumes that both α and β are constant over the entire sample is

Rt = α + βRMt+ εt, t = 1, , T

εt ∼ iid N(0, σ2) independent of RM t There are three main cases of interest: (1) β may differ over the two subsamples; (2) α may differ over the two subsamples; (3) α and β may differ over the two subsamples 1.5.1 Testing Structural Change in β only

If α is the same but β is different over the subsamples then we really have two market model regressions

Rt = α + β1RM t+ εt, t = 1, , TB

Rt = α + β2RM t+ εt, t = TB+1, , T that share the same intercept α but have different slopes β1 6= β2 We can capture such a model very easily using a step dummy variable defined as

Dt = 0, t≤ TB

= 1, t > TB

and re-writing the MM regression as the multiple regression

Rt= α + βRM t+ DtRMt+ εt The model for the first subsample when Dt= 0 is

Rt= α + βRM t+ εt, t = 1, , TB

and the model for the second subsample when Dt= 1 is

Rt = α + βRM t+ δRMt+ εt, t = TB+1, , T

= α + (β + δ)RM t+ εt Notice that the “beta” in the first sample is β1 = β and the beta in the second subsample is β2 = β + δ If δ < 0 the second sample beta is smaller than the first sample beta and if δ > 0 the beta is larger

We can test the constancy of beta over time by testing δ = 0:

H0 : (beta is constant over two sub-samples) δ = 0 vs H1 : (beta is not constant over two sub-samples The test statistic is simply the t-statistic

tδ=0 = bδ− 0

d

SE(bδ) =

b

δ

d

SE(bδ)

and we reject the hypothesis δ = 0 at the 5% level, say, if|tδ=0| > tT −3(0.025)

Example 4 MM regression for IBM cont’d

Consider the estimated MM regression equation for IBM using ten years of monthly

data from January 1978 through December 1987 We want to know if the beta on

IBM is using the first five years of data (January 1978 - December 1982) is different

from the beta on IBM using the second five years of data (January 1983 - December

1987) We define the step dummy variable

Dt = 1 if t > December 1982

= 0, otherwise The estimated (unrestricted) model allowing for structural change in β is given by

d

RIBM,t = −0.0001

(0.0045) + 0.3388

(0.0837) ·RM,t+ 0.3158

(0.1366) ·Dt· RM,t,

R2 = 0.311, σb ε = 0.0496

The estimated value of β is 0.3388, with a standard error of 0.0837, and the estimated

value of δ is 0.3158, with a standard error of 0.1366 The t-statistic for testing δ = 0

is given by

tδ=0 = 0.3158

0.1366 = 2.312 which is greater than t117(0.025) = 1.98 so we reject the null hypothesis (at the

5% significance level) that beta is the same over the two subsamples The implied

estimate of beta over the period January 1983 - December 1987 is

ˆ

β + ˆδ = 0.3388 + 0.3158 = 0.6546

It appears that IBM has become more risky

1.5.2 Testing Structural Change in α and β

Now consider the case where both α and β are allowed to be different over the two subsamples:

Rt = α1+ β1RM t+ εt, t = 1, , TB

Rt = α2+ β2RM t+ εt, t = TB+1, , T The dummy variable specification in this case is

Rt= α + βRMt+ δ1· Dt+ δ2· DtRMt+ εt, t = 1, , T

When Dt= 0 the model becomes

Rt = α + βRM t+ εt, t = 1, , TB,

so that α1 = α and β1 = β, and when Dt= 1 the model is

Rt= (α + δ1) + (β + δ2)RM t+ εt, t = TB+1, , T,

so that α2 = α + δ1 and β2 = β + δ2 The hypothesis of no structural change is now

H0 : δ1 = 0 and δ2 = 0 vs H1 : δ1 6= 0 or δ2 6= 0 or δ1 6= 0 and δ2 6= 0

The test statistic for this joint hypothesis is the F-statistic

Fδ 1 =0,δ 2 =0 = (SSRR− SSRU R)/2

SSRU R/(T − 4) since there are two restrictions and four regression parameters estimated under the unrestricted model The unrestricted (UR) model is the dummy variable regression that allows the intercepts and slopes to differ in the two subsamples and the restricted model (R) is the regression where these parameters are constrained to be the same

in the two subsamples

The unrestricted error sum of squares, SSRU R, can be computed in two ways The first way is based on the dummy variable regression The second is based on estimating separate regression equations for the two subsamples and adding together the resulting error sum of squares Let SSR1 and SSR2 denote the error sum of squares from separate regressions Then

SSRU R= SSR1+ SSR2 Example 5 MM regression for IBM cont’d

The unrestricted regression is

d

RIBM,t = −0.0001

(0.0065) + 0.3388

(0.0845) ·RM t

+ 0.0002

(0.0092) ·Dt+ 0.3158

(0.1377) ·Dt· RM t,

R2 = 0.311, σb ε = 0.050, SSRU R = 0.288379, and the restricted regression is

d

RIBM,t = −0.0005

(0.0046) + 0.4568

(0.0675) ·RM t,

R2 = 0.279, σb ε = 0.051, SSRR= 0.301558

The F-statistic for testing H0 : δ1 = 0 and δ2 = 0 is

Fδ 1 =0,δ 2 =0 = (0.301558− 0.288379)/2

0.288379/116 = 2.651 The 95% quantile, F2,116(0.05), is approximately 3.07 Since Fδ 1 =0,δ 2 =0 = 2.651 < 3.07 = F2,116(0.05) we do not reject H0 : δ1 = 0 and δ2 = 0 at the 5% significance level It is interesting to note that when we allow both α and β to differ in the two subsamples we cannot reject the hypothesis that these parameters are the same between two samples but if we only allow β to differ between the two samples we can reject the hypothesis that β is the same

An interesting question regarding the beta of an asset concerns the stability of beta over the market cycle For example, consider the following situations Suppose that the beta of an asset is greater than 1 if the market is in an “up cycle”, RM t > 0, and less than 1 in a “down cycle”, RMt < 0 This would be a very desirable asset to hold since it accentuates positive market movements but down plays negative market movements We can investigate this hypothesis using a dummy variable as follows Define

Dtup = 1, RM t > 0

= 0, RM t ≤ 0

Then Dtupdivides the sample into “up market” movements and “down market” move-ments The regression that allows beta to differ depending on the market cycle is then

Rt = α + βRM t+ δDupt · RM t+ εt

In the down cycle, when Dtup= 0, the model is

Rt= α + βRM t+ εt

and β captures the down market beta, and in the up market, when Dupt = 1, the model is

Rt = α + (β + δ)RM t+ εt

so that β + δ capture the up market beta The hypothesis that β does not vary over the market cycle is

and can be tested with the simple t-statistic tδ=0 = b δ−0

c

SE( b δ)

If the estimated value of δ is found to be statistically greater than zero we might then want to go on to test the hypothesis that the up market beta is greater than one Since the up market beta is equal to β + δ this corresponds to testing

H0 : β + δ = 1 vs H1 : β + δ ≥ 1 which can be tested using the t-statistic

tβ+δ=1 = β +b bδ− 1

d

SE(β +b bδ).

Since this is a one-sided test we will reject the null hypothesis at the 5% level if

tβ+δ=1 <−t0.05,T −3

Example 6 MM regression for IBM and DEC

For IBM the CAPM regression allowing β to vary over the market cycle (1978.01

- 1982.12) is

d

RIBM,t = −0.0019

(0.0111) + 0.3163

(0.1476) ·RM t+ 0.0552

(0.2860) ·Dupt · RM t

R2 = 0.201,σ = 0.053b

Notice that bδ = 0.0552, with a standard error of 0.2860, is close to zero and not

estimated very precisely Consequently, tδ=0 = 0.05520.2860 = 0.1929 is not significant at any reasonable significance level and we therefore reject the hypothesis that beta varies over the market cycle However, the results are very different for DEC (Digital Electronics):

d

RDEC,t = −0.0248

(0.0134)

+ 0.3689

(0.1779) ·RM t+ 0.8227

(0.3446) ·Dupt · RM t

R2 = 0.460,σ = 0.064.b

Here bδ = 0.8227, with a standard error of 0.3446, is statistically different from zero

at the 5% level since tδ=0 = 2.388 The estimate of the down market beta is 0.3689, which is less than one, and the up market beta is 0.3689 + 0.8227 = 1.1916, which